The present invention relates generally to coding techniques for communications and, more particularly, to a low-complexity high-performance low-rate communications code.
Low-rate codes play a critical role for communication systems operating in the low signal-to-noise ratio (SNR) regime (e.g., code-spread communication systems and power-limited sensor networks). The ultimate Shannon capacity of an additive white Gaussian noise (AWGN) channel in terms of signal-to-noise ratio (SNR) per information bit is about −1.6 dB for codes with rates approaching zero. Hadamard codes and super-orthogonal convolutional codes are traditional low-rate channel coding schemes which offer low coding gain and hence their performance is far away from the Shannon limit. As shown by the block diagram 10B of FIG. 1B, Turbo Hadamard codes are parallel concatenated codes and the component codes are convolutional-Hadamard codes 11B1, 11B2 and 11BM. Normally, the number of the states of the convolutional-Hadamard codes is larger than 2
The repeat-accumulate (RA) codes have proved to achieve the Shannon limit in AWGN channels when the coding rate approaches zero. As shown by the block diagram 10A in FIG. 1A of an RA code encoder with two channel outputs 13A, 14A, the outer code is a repetition code 11A interconnected by an interleaver 15A to the inner code 12A which is a rate-one accumulator. For a low density parity check (LDPC) code, the inner code is a single parity-check code. Note that in the low-rate regime, they will suffer large performance loss from the simulation, although they can approach capacity from the DE analysis and optimization. For instance, for LDPC code of rate 0.05, the simulated SNR threshold is larger than 0 dB which is far away from the capacity and much worse than RZH code with the same rate.
The repeat-accumulate (RA) codes and low-density parity-check (LDPC) codes have capacity-approaching capability for various code rates. However, in the low-rate region, both codes suffer from performance loss and extremely slow convergence speed with iterative decoding. Low-rate turbo-Hadamard codes offer good performance with faster convergence speed than RA codes and LDPC codes with the help of Hadamard code words. However, for turbo-Hadamard codes to perform well a relatively large Hadamard order is normally required which increases the complexity of the decoder. Accordingly, there is a need for low-rate code with a similar performance as a turbo-Hadamard code and with lower complexity.